The P-Norm

Feynman's functional integral averages a "contribution formula", or path integrand, over all possible paths a particle can take in space-time. The contribution a path makes to the functional integral is proportional to exp(iS), with S the action along the trajectory. For an integer value of the time-slicing (discretization) index, the standard formulation, the action S is a real quantity. In QCI, however, with a complex value for the index, S itself also becomes complex.

Feynman's original formulation only defines the contribution formula for real S. Thus, any extension of this contribution formula to complex time-slicing index seems equally valid, provided that it reduces to exp(iS) for real S. For example, exp(iS*), exp(i Re(S)), and exp(i |S|) are all equally valid ways to define the contribution formula for complex time-slicing index, as they are consistent with the known formula exp(iS) for real values of that index. Indeed, our universe, in which all experiments have been performed, corresponds to a real (or close to real) value of the index, in the QCI model.

In Feynman's original interpretation, only the phase of the contribution changes from path to path, while its magnitude remains constant. That is, the contribution formula is a pure phase factor, depending upon the action S. It seems reasonable, while extending the formula to complex index, to maintain this fundamental property. Thus, we should use a general formula of the form exp(i norm(S)), taking a norm of the path action, in order to obtain an extension of the path integrand consistent with the phase property. If we extended with the obvious formula given by exp(i S), the contribution would no longer be a pure phase.

We chose the p-norm for this purpose, as this class of norms are the simplest and most important on the complex numbers, including the Euclidean, Manhattan, and Maximum norms. As such, each different value of p corresponds to a different, but equally consistent way to extend the contribution factor to complex values of the index. We are also open to considering different norms as well.